e et aL. Enhanced fracture toughness by ceramic laminate desig Rao et al. have demonstrated how a rearrangement -0.19277, A21=2. 55863, A 22=-12-6415, 423=197630, of equation (4)can lead to a threshold strength criterion A24=-10 9860 for ceramic laminates. This finding has significant During crack propagation in ceramic materials, an implications for the statistical strength distribution of interaction zone develops behind the crack front. within ceramic components. The threshold strength phenom- this interaction zone, the two crack surfaces are not enon exists as failure can only occur at an applied stress completely separated, leading to some crack-surface sufficient to drive the crack through the compressive interactions. These interactions are capable of trans- layers. Using this assumption, and by setting the crack itting stresses due to grain/grain friction, or via grain length a=(t2+ 2t1)/2 and K=Kie(the critical stress crack bridging. These types of interactions are respon- intensity factor for the compressive material), equation sible for R curve behaviour in monolithic ceramic 4)may be rearranged to produce an expression for materials. In the remaining discussion, the effect of threshold strength, i. e. the strength below which no both types of crack-surface interaction will be termed failure is possible ridging stresses abr. The stepwise residual stress pattern also acts in this interaction zone behind the propagating crack as well as at the crack tip. Compressive stresses (1+ behind the propagating crack act to close the crack whilst tensile stress regions act as further drivers for pening the crack. The stress intensity factor at the crack tip itself can therefore be deter principle of superposition, by summing the contribu Threshold strength is therefore dependent upon two tions of the applied stress Ka, the bridging stress Kbr and main factors for a given material system, the compres- the residual stress Kr sive stress and the ratio of tensile layer thickness to compressive layer thickness. To obtain high threshold Ka+Kr+kbr (11) streng th, the compressive stress should naturally Crack tip extension(measured by KR)is predicted to stress, the laminate architecture should be designed to Ko. For a crack with initial crack length ao, 4.ughness maximised. However, for a given maximum compressive occur when Ktip equ juals the intrinsic crack tip be compressive layers, thereby maximising the t lt ratio. laminate. as the crack (length a) advances through incorporate very small separations between successive The threshold strength phenomenon therefore trun- cates the statistical strength distribution, yielding an KR(a, o)=Ko(a)-Kr(a)-kbr(a, ao) (12) apparently high Weibull modulus material. Howeve If Ko, or and Obr are known(or can be calculated with this threshold strength calculation is only valid for sufficient precision), it becomes possible to predict(as a cracks that propagate straight through the compressive function of a)the R curve behaviour of a laminated layer without bifurcation or significant deflection. In structure. Lakshminaravanan et al.2223 and Moon such cases, where bifurcation is exhibited by the crack as et al.24, 25 have demonstrated how this can be used to it propagates through the compressive layer, a greater redict the fracture behaviour of layered ceramics threshold strength may be observed than predicted by the model. Quantitative prediction of threshold strength addition, weight function analysis makes it possible to in such cases is extremely complicated determine the separate effects of the residual stress Fracture mechanics weight function analysis distribution and the crack bridging closure stresses on Bueckner'9 first demonstrated that the stress intensity the measured R curve factor for an edge crack of depth a can be calculated by The present study considers the behaviour of multi weight function h(x a)(dependent upon geometry) and crack bifurcation. The use of weight function analysis any stress distribution a(x)acting normal to the fracture for modelling the apparent fracture toughness of plane(x is the distance along the crack measured from ceramic laminates is discussed with reference to optimis- ing the macrostuctural design. A laminate design oncept is presented that incorporates macrostructural K= h(x, a)o(x)d: 9 Constraints due to the matrix cracking phenomena, requirements for bifurcation, ASTM test bar geometry A suitable weight function for a single edge V-notched and the additional stress incurred during ceramic beam(SEVNB) sample and notch geometry has been achining. given by Fett and munz h(x,=/2)2 Experimental details (1-)2(1-#) Si3N4, Si3 Na-TiN and Si3N4-TiB, ceramics were pre- pared using commercially available a-Si3N4 powder (FCT Technology GmbH, Germany) with 5 wt-%Y2O3 (10)and 2 wt-%Al203 sintering aids and TiB2(H. C Starck, de F)or TIN (H. C. Starck). The required composi- using the following values for the coefficients Ayu tion was then ball milled in isopropanol for 5 h using (Ref.20):A0o=0.4980,Ao1=24463,A2=0·0700,A Si3Na milling media. Powders for roll compaction were 87, A04=-3-067, A10=0.54165, A1l=-5-0806, prepared by adding 4 wt-% crude rubber (plasticiser) A12=243447, A13=-32-7208, A14=181214, A20= with 3 wt-% petrol (solvent)to a mixture of the powders. 106 Advances in Applied Ceramics 2005 VOL 104 No 3Rao et al.9 have demonstrated how a rearrangement of equation (4) can lead to a threshold strength criterion for ceramic laminates. This finding has significant implications for the statistical strength distribution of ceramic components. The threshold strength phenom￾enon exists as failure can only occur at an applied stress sufficient to drive the crack through the compressive layers. Using this assumption, and by setting the crack length a5(t2z2t1)/2 and K5KIc (the critical stress intensity factor for the compressive material), equation (4) may be rearranged to produce an expression for threshold strength, i.e. the strength below which no failure is possible: sthr~ KIc p t2 2 1z 2t1 t2 h i
1=2 zs1 1{ 1z t1 t2  2 p sin{1 1 1z2t1 t2 " # ! (8) Threshold strength is therefore dependent upon two main factors for a given material system, the compres￾sive stress and the ratio of tensile layer thickness to compressive layer thickness. To obtain high threshold strength, the compressive stress should naturally be maximised. However, for a given maximum compressive stress, the laminate architecture should be designed to incorporate very small separations between successive compressive layers, thereby maximising the t1/t2 ratio. The threshold strength phenomenon therefore trun￾cates the statistical strength distribution, yielding an apparently high Weibull modulus material. However, this threshold strength calculation is only valid for cracks that propagate straight through the compressive layer without bifurcation or significant deflection. In such cases, where bifurcation is exhibited by the crack as it propagates through the compressive layer, a greater threshold strength may be observed than predicted by the model. Quantitative prediction of threshold strength in such cases is extremely complicated. Fracture mechanics weight function analysis Bueckner19 first demonstrated that the stress intensity factor for an edge crack of depth a can be calculated by integrating over the entire crack length, the product of a weight function h(x,a) (dependent upon geometry) and any stress distribution s(x) acting normal to the fracture plane (x is the distance along the crack measured from the surface): K~ ða 0 h(x,a)s(x)dx (9) A suitable weight function for a single edge V-notched beam (SEVNB) sample and notch geometry has been given by Fett and Munz:20 h(x,a)~ 2 pa 1=2 1 1{ x a  1=2 1{ a W  3=2 1{ a w
3=2 zXAnm 1{ x a
nz1 a W
m  (10) using the following values for the coefficients Anm (Ref. 20): A0050.4980, A0152.4463, A0250.0700, A035 1.3187, A04523.067, A1050.54165, A11525.0806, A12524.3447, A135232.7208, A14518.1214, A205 20.19277, A2152.55863, A225212.6415, A23519.7630, A245210.9860. During crack propagation in ceramic materials, an interaction zone develops behind the crack front. Within this interaction zone, the two crack surfaces are not completely separated, leading to some crack–surface interactions.21 These interactions are capable of trans￾mitting stresses due to grain/grain friction, or via grain crack bridging. These types of interactions are respon￾sible for R curve behaviour in monolithic ceramic materials. In the remaining discussion, the effect of both types of crack–surface interaction will be termed bridging stresses sbr. The stepwise residual stress pattern also acts in this interaction zone behind the propagating crack as well as at the crack tip. Compressive stresses behind the propagating crack act to close the crack whilst tensile stress regions act as further drivers for opening the crack. The stress intensity factor at the crack tip itself can therefore be determined, using the principle of superposition, by summing the contribu￾tions of the applied stress Ka, the bridging stress Kbr and the residual stress Kr: Ktip~KazKrzKbr (11) Crack tip extension (measured by KR) is predicted to occur when Ktip equals the intrinsic crack tip toughness K0. For a crack with initial crack length a0, KR can be predicted as the crack (length a) advances through the laminate: KR(a,a0)~K0(a){Kr(a){Kbr(a,a0) (12) If K0, sr and sbr are known (or can be calculated with sufficient precision), it becomes possible to predict (as a function of a) the R curve behaviour of a laminated structure. Lakshminarayanan et al.22,23 and Moon et al.24,25 have demonstrated how this can be used to predict the fracture behaviour of layered ceramics containing macroscopic residual stress patterns. In addition, weight function analysis makes it possible to determine the separate effects of the residual stress distribution and the crack bridging closure stresses on the measured R curve. The present study considers the behaviour of multi￾layer TiN–Si3N4 laminar composites that did not exhibit crack bifurcation. The use of weight function analysis for modelling the apparent fracture toughness of ceramic laminates is discussed with reference to optimis￾ing the macrostuctural design. A laminate design concept is presented that incorporates macrostructural constraints due to the matrix cracking phenomena, requirements for bifurcation, ASTM test bar geometry and the additional stress incurred during ceramic machining. Experimental details Si3N4, Si3N4–TiN and Si3N4–TiB2 ceramics were pre￾pared using commercially available a-Si3N4 powder (FCT Technology GmbH, Germany) with 5 wt-%Y2O3 and 2 wt-%Al2O3 sintering aids and TiB2 (H. C. Starck, grade F) or TiN (H. C. Starck). The required composi￾tion was then ball milled in isopropanol for 5 h using Si3N4 milling media. Powders for roll compaction were prepared by adding 4 wt-% crude rubber (plasticiser) with 3 wt-% petrol (solvent) to a mixture of the powders. Gee et al. Enhanced fracture toughness by ceramic laminate design 106 Advances in Applied Ceramics 2005 VOL 104 NO 3